Question

in 2012, the population of a small town was 3,550. the population is decreasing at a rate of 2.9% per year. how can you rewrite an exponential decay function to find the quarterly rate?
○ d. multiply the annual growth rate by $\frac{1}{4}$ so it is a quarter of the rate.
what is the quarterly decay rate?
the quarterly decay rate is %.
(type an integer or decimal rounded to three decimal places as needed.)

Explanation:

Step1: Identify annual decay rate

Annual decay rate $r_{annual} = 2.9\% = 0.029$

Step2: Calculate quarterly decay rate

Divide annual rate by 4:
$r_{quarterly} = \frac{0.029}{4}$

Step3: Convert to percentage

Multiply result by 100:
$r_{quarterly (\%)} = \frac{0.029}{4} \times 100$

Answer:

Method to rewrite the function:

To write an exponential decay function for quarterly decay, use the form $P(t) = P_0(1 - r_{quarterly})^{4t}$, where $P_0=3550$, $r_{quarterly}$ is the quarterly decay rate in decimal form, and $t$ is time in years. This replaces the annual decay form $P(t)=P_0(1 - 0.029)^t$ by using the quarterly rate and adjusting the exponent to account for 4 quarters per year.

Quarterly decay rate:

0.725%